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Solution - Absolute value equations

Exact form:

a = 0

a=0

See steps

Step-by-step explanation

1. Rewrite the equation without absolute value bars

Use the rules:
$| x | = | y |$ → $x = \pm y$ and $| x | = | y |$ → $\pm x = y$
to write all four options of the equation
$| a + 1 | = | a - 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| a + 1 \| = \| a - 1 \|$
$x = + y$	$(a + 1) = (a - 1)$
$x = - y$	$(a + 1) = - (a - 1)$
$+ x = y$	$(a + 1) = (a - 1)$
$- x = y$	$- (a + 1) = (a - 1)$

When simplified, equations $x = + y$ and $+ x = y$ are the same and equations $x = - y$ and $- x = y$ are the same, so we end up with only 2 equations:

$\| x \| = \| y \|$	$\| a + 1 \| = \| a - 1 \|$
$x = + y$ , $+ x = y$	$(a + 1) = (a - 1)$
$x = - y$ , $- x = y$	$(a + 1) = - (a - 1)$

2. Solve the two equations for a

5 additional steps

$(a + 1) = (a - 1)$

Subtract from both sides:

$(a + 1) - a = (a - 1) - a$

Group like terms:

$(a - a) + 1 = (a - 1) - a$

Simplify the arithmetic:

$1 = (a - 1) - a$

Group like terms:

$1 = (a - a) - 1$

Simplify the arithmetic:

$1 = - 1$

The statement is false:

$1 = - 1$

The equation is false so it has no solution.

9 additional steps

$(a + 1) = - (a - 1)$

Expand the parentheses:

$(a + 1) = - a + 1$

Add to both sides:

$(a + 1) + a = (- a + 1) + a$

Group like terms:

$(a + a) + 1 = (- a + 1) + a$

Simplify the arithmetic:

$2 a + 1 = (- a + 1) + a$

Group like terms:

$2 a + 1 = (- a + a) + 1$

Simplify the arithmetic:

$2 a + 1 = 1$

Subtract from both sides:

$(2 a + 1) - 1 = 1 - 1$

Simplify the arithmetic:

$2 a = 1 - 1$

Simplify the arithmetic:

$2 a = 0$

Divide both sides by the coefficient:

$a = 0$

3. Graph

Each line represents the function of one side of the equation:
$y = | a + 1 |$
$y = | a - 1 |$
The equation is true where the two lines cross.

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Why learn this

We encounter absolute values almost daily. For example: If you walk 3 miles to school, do you also walk minus 3 miles when you go back home? The answer is no because distances use absolute value. The absolute value of the distance between home and school is 3 miles, there or back.
In short, absolute values help us deal with concepts like distance, ranges of possible values, and deviation from a set value.

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for a

3. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for a

3. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved